This is more the classical crank canard than anything. "It's not me being unreasonably hostile and dogmatic against established beliefs, it's established beliefs being unreasonably hostile and dogmatic towards my beliefs!"

Having read a book on the history of cranks and crank psychology, the pattern of defense is all too familiar.

Set theory "works" more often because it is more powerful. In computer science, a language being less powerful is a feature. If you read the screencap I posted from Baez & Stay, you'll see how intuitionist logic makes distinctions that classical logic conflates.

What do you think I am defending? Because I'm pretty sure that I don't even care in the way you seem to think I do.

I'm honestly not sure, but the consistent strain is that modern mathematics is doing something "wrong", and either needs to be or will be corrected.

f[sup]2[/sup](x) = f(f(x))

sin[sup]2[/sup](x) = sin(x)[sup]2[/sup]

Goddamnit yeah, that one I can't defend. It's even worse that sin^{-1}(x) isn't 1/sin(x).

Even worse, in measure theorhetic terms, we use f^2 to mean f·f, but once you claim we're using the Lebesgue measure and f is Riemann integrable, f^2 is not f^2(x).

It's all a sham.

I can't say I care a whole lot, but somebody whose taste in mathematics I trust is Yuri Manin. I noticed his name in the Wikipedia article for "actual infinity", and the following paper cited there seems to contain a lot of interesting looking criticisms.

https://arxiv.org/pdf/math/0209244.pdf

I read a section here:



I am in total, absolute agreement about his notion that category theory fixes any issues with working with large collections. Where I'm perplexed is how this amounts to a criticism of mathematics at all - it sounds to me like a strong praise of the work of Grothendieck, who I think many will recognize as the best mathematician of the latter half of the 20th century.

The thing is, we already work in this world. Nobody takes set theory in the naive sense commented here, and for anyone where such topics are pertinent, we study and understand enough category theory to understand these things. Do you think any pure math departments aren't deeply influenced by the advent of category theory in this way?

I'm already aware of different categories, and their usefulness in describing sort of meta-mathematical collections that can't be described by set theory, and I don't even know very much.



Here he's also basically calling finitism, intuitionism and these complaints about set theory unimportant, his claim is that category theory gives us language to talk about the places where set theory doesn't work so well. Fine! That's something we already live by, in fact that means we don't have cause to "cut back" set theory any further.

Oh, I totally agree that finitism is unimportant.

Actually, I would love to understand set theory well enough to believe that it is crucially important. Well, sort of. If I have time. Actually, if I ever do that, it will probably be by reading Manin's book, A Course in Mathematical Logic for Mathematicians. It's just that such an undertaking really is quite philosophical and ambitious in order to be done tastefully in my mind, because the field seems to be so chock full of epic battles that are now only of historical importance.

I like the following passage:

[quote=Yuri Manin]
Once this view was shown to be only an approximation to the incomparably more sophisticated quantum description, sets lost their direct roots in reality. In fact, the structured sets of modern mathematics used most effectively in modern physics are not sets of things, but rather sets of possibilities. For example, the phase space of a classical mechanical system consists of pairs (coordinate, momentum) describing all possible states of the system, whereas after quantization it is replaced by the space of complex probability amplitudes: the Hilbert space of L2–functions of the coordinates or something along these lines. The amplitudes are all possible quantum superpositions of all possible classical states. It is a far cry from a set of things.

Moreover, requirements of quantum physics considerably heightened the degree of tolerance of mathematicians to the imprecise but highly stimulating theoretical discourse conducted by physicists. This led, in particular, to the emergence of Feynman’s path integral as one of the most active areas of research in topology and algebraic geometry, even though the mathematical status of path integral is in no better shape than that of Riemann integral before Kepler’s “Stereometry of Wine Barrels”.

Computer science added a much needed touch of practical relevance to the essentially hygienic prescriptions of formal logic. The introduction of the notion of “success with high probability” into the study of algorithmic solvability helped to further demolish mental barriers which fenced off foundations of mathematics from mathematics itself.
[/quote]

This is what I am talking about: should I feel like I'm not wasting my time if I wade through dated philosophical debates about the meaning of things like Godel's incompleteness theorem, only to later see that it is sort of a trivial technical result?

[quote=Yuri Manin]
Insofar as it is accepted in thousands of research papers, one can simply say that the language of mathematics is the language of set theory.

Since the latter is so easily formalized, this allowed logicists to defend the position that their normative principles should be applied to all of mathematics and to overstate the role of “paradoxes of infinity” and G ̈odel’s incompleteness results.

However, this fact also made possible such self–reflexive acts as inclusion of metamathematics into mathematics, in the form of model theory. Model theory studies special algebraic structures – formal languages – considered in turn as mathematical objects (structured sets with composition laws, marked elements etc.), and their interpretations in sets. Baffling discoveries such as G ̈odel’s incompleteness of arithmetics lose some of their mystery once one comes to understand their content as a statement that a certain algebraic structure simply is not finitely generated with respect to the allowed composition laws.
[/quote]

If set theory is shown to be unimportant, that's going to take a ton of work. For ask much as we can do with category theory in studies that work nicely with it, like algebraic topology, analysis has so far not found any reason to make use of it, and is founded pretty much solely in set theory. So changing how we use sets will have to change how basically all of analysis is treated. No small task!

I'm not even sure what a set theory free conception of measure theory could even look like.



Once you learn Godel's incompleteness theorem is a sort of trivial technical result, you've definitely seen something very important. It will help you realize how much the types of "deep philosophical discussions" people have about it are really dumb.

But yeah, nobody demands you slog through old stuff about math foundations. Why would you think that?

In my experience, a decent way to understand a topic is to look for an exposition that is sensitive to its historical development, because it makes for a more honest and illuminating understanding of the topic. It's just that when one tries to do this with set theory, well, you kind of get the opposite experience, precisely because the development of the subject was so traumatic. And this despite the obvious reality that so much of set theory was in fact motivated by (confused?) philosophical interpretations in the first place!

Don't get me wrong, those early papers really are interesting. It's just that the spirit in which they are written probably doesn't have much to do with math. Actually, my understanding is that they actually have more to do with computer science.

This is probably one of the reasons why finitists like Zeilberger don't even like continuous analysis, and prefer to think about its discrete analogues as more "meaningful".

Isn't the whole point of this subthread that I actually prefer new math foundations?

Yes, I looked over the ancient set theory motivated stuff before. It's ugly looking and apparently mostly useless except in ways that have already been subsumed into the Bourbaki tradition of mathematics that we all accept.

For example, is there anything gained by learning about axiomatic set theory in more detail than is necessary for a measure theory class?

I thought that because it was implicit in what Paul Taylor says the logicians presume me to accept, simply by using and feigning understanding mathematics as accepted by the mainstream Bourbakist tradition.

Don't get me wrong, I think classical mathematics is beautiful and indispensable (you mentioned pretty much all of real analysis), and I own plenty 'Bourbakist' books that I'd like to read someday.

well, that would certainly make integration easier.

[quote=Paul Taylor]
Ever since Logic became a mathematical discipline in the nineteenth century, foundational discussions (whether using sets, types or categories) have been based on four premises:

1. logicians know best, and mathematicians should be grateful for what they are given;
2. it is up to mathematicians to glue the continuum back together from the dust that
   logicians have provided; whilst
3. they have convinced each other that foundations should permit arbitrary abstraction of mathematical processes; and
4. anyone who tinkers with the foundations risks bringing the whole edifice of science crashing down.

[/quote]

.

Maybe, but you might need Maple.

I haven't paid a lot of attention to this discussion. My position is basically this: mathematics is a technology. We are free to do mathematics however we want. All axioms, including those of set theory, were chosen because they are immediately convenient for us, not because they reveal some sort of foundational Truth from God. There's no particular reason that everybody needs to do mathematics from the same set of axioms, or to discourage people who want to rebase mathematics on axioms that are more personally convenient for them.

It sounds like finitists are effectively trying to reconstruct mathematics in terms of what is possible on a digital computer. That seems like incredibly valuable and interesting work to me.

Maybe that's part of the difference. I'm looking at a version of mathematics which has solved these problems, not a version which is still questioning these things. So I don't see the same problems because they don't really exist.



Bourbaki is dead. It doesn't have influence on how mathematics is done today. People do not write textbooks in that fashion anymore.

Study ZF axioms if you're curious about ZF axioms. Knowing how they work is important up to a point, obviously a deep analysis is unnecessary. I haven't slogged through the debates, I've just read enough to know what I need to, and enough of the debates to feel confident why we use what we use. But analysts like it, and so far, there's no problems with them using it, and moreover it's highly unlikely that people will want to rework the massive portfolio of analysis we have, so short of a contradiction in ZF that's massively painful to work through, some sort of axiomatic treatment of set theory is going to be with us forever.

Yeah, expanding on type theory and category theory foundations, and more finitist math are all good things. The only issue I take is there is a strong overlap of people who believe in finitism and people who are broadly critical of the entire mathematical tradition. It's not good to discourage people who want to rebase math, but it's also not good to be senselessly critical of everyone else.

Play nice, people. That's all I want.

Oh yeah, I forgot about that time I was discussing ZF last week and we paused to remember that logicians know best..

Just to be clear, I am not talking about reading debates. I am talking about reading papers which implicitly presume such debates (wrongly in my mind) to be important. In other words, it is hard to read original papers in set theory and logic, because they don't have the benefit of hindsight. Of course the same can be said of most research, but as far as mathematics goes, I find set theory and logic to be particularly inaccessible.

Sure, I could read a standard textbook on ZF set theory. Or I could read Yuri Manin's book, A Course in Mathematical Logic for Mathematicians, and find out how the sausage is really made. This is a book that not only subsumes modern insights, but also doesn't shy away from the problems with the whole enterprise in the first place.

Just to be clear, the continuing influence of Bourbaki is historical, in the sense that it was the idea of Bourbaki in the first place to apply set theory so systematically. Changing the style of the very same material as written in modern textbooks doesn't change this.

As I've said, I own several "Bourbakist" books (basically, anything that relies heavily on measure theory, integration, or even functional analysis), and I think the content is fantastic.

Probably because set theory, in the philosophical sense, is a much bigger and more complicated area than the more limited version we use in mathematics, as is logic. They're naturally inaccessible because normally people take a few years of classes in logic and philosophy before getting there.



Or you could just study math, which is what mathematicians are doing. I guess, though, if you're really into logic, and you want to know that, and study that, great! I'm all the happier for you, because that stuff is really cool. But that stuff is reallllly not super important to mathematicians.



Ah, yeah, in that vein you're right.

fixed

The book I've been mentioning is literally called A Course in Mathematical Logic for Mathematicians, and was written by an expert in algebraic geometry, computation, and physics, and is a co-inventor of quantum computation. I am pretty sure that counts as "what mathematicians are doing"!



This is humorously incorrect. You seem to be oblivious of a number of rather deep threads that run between logic, computation, and physics.

No, I am talking about original papers in mathematical logic. Maybe it's just an impedance mismatch with my intuitive understanding of how to think about math, or I never really learned logic properly, but I find the papers pretty darn dense.

Honestly, at this point I'd rather spend my time doing stuff in Haskell or Prolog than try to understand what somebody in 1940 thought was important. As far as I'm concerned, the useful parts have been subsumed into computer science and type theory (but maybe that's just divergent motivations on my part, since I'd be more inclined to sink 4 hours into learning a logical construct in a programming language than in trying to understand some ancient paper).

Also, if you peruse Manin's book, you will find a ton of "applied" axiomatic set theory, ranging from problems of solvability of Diophantine equations, quantum groups, P vs. NP, Church's thesis in the language of category theory, and so on. Really, it's quite a rich subject, and has more to do with computer science than real analysis. It's just that this heritage of set theory, while useful for computer science (and a limited number of applications to classical mathematics, like the solvability of Diophantine equations), is kind of an albatross around the neck of classical mathematicians, save for instances in which it really is the right language (like real analysis). And that is the motivation for creating cleaner foundations like Paul Taylor is working on, which sidestep the whole mistake of trying to formalize vauge, intuitive things like the "continuum".

I guess the philosophical part arises when, if Paul Taylor has to basically ignore real analysis, and we admit that a lot of real analysis is based on formalizing intuitive things like the continuum, then: how much of real analysis really is representative of the physical world? Certainly, when physicists do things like integration (even when mathematicians struggle to interpret it rigorously), this definitely says that calculus has something to do with how the universe works.

Maybe Andrej Bauer put it better than I can.

[quote=Andrej Bauer]
It seems to me that people think I am a constructive mathematician, or worse a constructivist (a word which carries a certain amount of philosophical stigma). Let me be perfectly clear: it is not decidable whether I am a constructive mathematician.

But seriously, if anything, you may call me a mathematical relativist: there are many worlds of mathematics, and the view of the worlds is relative to which one I am in. Any attempt to bring mathematics within the scope of a single foundation necessarily limits mathematics in unacceptable ways. A mathematician who sticks to just one mathematical world (probably because of his education) is a bit like a geometer who only knows Euclidean geometry. This holds equally well for classical mathematicians, who are not willing to give up their precious law of excluded middle, and for Bishop-style mathematicians, who pursue the noble cause of not opposing anyone.

What could be more appealing to a mathematician than the idea that there is not one, but many, infinitely many worlds of mathematics? Would he not want to visit them all, understand how they are related, and see what happens to his favorite subject as he moves between them?

Let us consider an example. The real numbers are a mathematical object of fundamental importance, and have many aspects:

1. The reals as a set are uncountable and in bijection with the powerset of natural numbers.
2. The reals as an algebraic structure form a linearly ordered field.
3. The reals as a space are locally compact, Hausdorff, and connected.
4. The reals are a measurable space on which measure theory rests.
5. The reals of non-standard analysis contain infinitesimals.
6. The reals as understood by Leibniz contain nilpotent infinitesimals.
7. The reals as Brouwerian continuum cannot be decomposed into two disjoint inhabited subsets.
8. The reals are overt.

We can have some of these properties but not all at once. History has chosen for us a combination that is taught today as a dogma. Any attempt to deviate from it is met with opposition. Thus you probably consider 1, 2, 3, and 4 as true, 5 as something exotic you heard of, 6 as Leibniz’s biggest mistake, 7 as intuitionistic hallucination (because obviously the reals can be decomposed into the non-negative and negative numbers), and 8 as something you never heard of (but you should have because it is the concept dual to compactness and you have been using it all your life).

Once we break free from Cantor’s paradise that Hilbert threw us in we discover unsuspected possibilities:

1. It may happen that the reals are in 1-1 correspondence with a subset of the natural numbers, while at the same time they form an uncountable set.
2. It may happen that the reals form a proper class.
3. It may happen that every real number has a Turing machine computing its digits.
4. It may happen that the reals are not linearly ordered.
5. It may happen that the reals are locally non-compact, in the sense that every interval contains a sequence without an accumulation point.
6. It may happen that every subset of the reals is measurable.
7. It may happen that the reals can be covered by a sequence of intervals whose cumulative length does not exceed 1.
8. It may happen that the reals contain nilpotent infinitesimals, which validate the 17th century calculations that physicists still use because, luckily, they did not subscribe entirely to the ϵδ-dogma of analysis.
9. It may happen that every real function is continuous, and consequently the reals are not decomposable into two disjoint inhabited subsets.
10. It may happen that the reals are not overt, whatever that means.

It should be admitted that some of the possibilities are rather bizarre. For example, I do not know what good it is to have the reals as a subset of the naturals, but I am sure somebody could think of something. But why should a measure theorist ignore a world of mathematics in which every subset is measurable, or a computer scientist one in which all reals are computable, or a topologist one in which all functions are continuous, or an analyst one in which all functions are smooth?

I am not proposing that mathematics should be compartmentalized so that each branch sits in its world of mathematics, incompatible with others. That would be a grave mistake indeed. In fact, the unification of mathematics under the umbrella of classical set theory has been immensely successful precisely because it allowed mathematicians to discover deep and unsuspected connections between different branches of mathematics. We have learnt to look for connections between branches of mathematics, and now we must also learn to look for connections that span worlds of mathematics.

We cannot ignore the many worlds of mathematics. Therefore, mathematics must become applicable in a wide variety of worlds. Mathematicians have to be educated so that they develop multiple mathematical intuitions that help them feel how the worlds of mathematics behave.

I have so far not given you any technical definition of a mathematical world. Such a definition may be useful for showing meta-theorems, but I think it can never be exhaustive. A world of mathematics may be a forcing extension of set theory, or a topos, or a pretopos, or a model of type theory, or any other structure within which it is possible to interpret the basic language of mathematics.

At the moment I am visiting the Institute for Advanced Study as a member of the Univalent Foundations group. We are building a new foundation of mathematics whose language is type theory rather than set theory, and whose primary objects are homotopy types and not just bare sets. Do I think this is an exciting new development? Certainly! Will the Univalent foundations disrupt the monopoly of Set-theoretic foundations? I certainly hope so! Will it become the new monopoly? It must not!
[/quote]

http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/

Dear god, you are relentless.

Yes, some mathematicians do stuff in that area. Baez is also well outside the norm of what typical mathematicians do.

Other than that I honestly don't have the patience to read through whatever stuff you're doing. Do what you like, but I'm not going to take these half-baked criticisms you throw out to heart.

Question: do you think I want to argue with you? It's OK for the two of us to have a difference of opinion, Reid! Try not to take it personally.

Anyway, it's not outside the norm at all from the point of view of mathematical physicists (look at Roger Penrose' work, for example). Have a look at that abstruse goose comic again and maybe get outside of your bubble a little.

Anyway, it would be easier to talk to you if you didn't take every single correction I made to your sloppy and condescending remarks as a crusade. I'm just trying to have a conversation, and I don't happen to agree with the way you are characterizing a lot of this stuff that you only seem to be dimly aware of. Hence the length of my posts.

I mean, really. I don't want to argue with you on these trivial things, but how else am I supposed to respond to this? This sentiment is just plain wrong, and it's not just because one or two mathematicians that I cited that work on it. "Not super important" "to mathematicians". Reid, you are a mathematician, and I gave you a counter-example to some sloppy reasoning. You are supposed to congratulate me, not lash out!

Is the part where you tell me that "details don't matter", and that I should calm down instead of correcting them, in order to see the big picture of whatever opinion it is you are trying to foist on me?